The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros
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We describe a number of results and techniques concerning the non-vanishing of automorphic L-functions at s = ½. In particular we show that as N → ∞ at least 50% of the values L(½, f), with f varying among the holomorphic new forms of a fixed even integral weight for Γ0(N) and whose functional equations are even, are positive. Furthermore, we show that any improvement of 50% is intimately connected to Landau-Siegel zeros. These results may also be used to show that X0(N) = Γ0(N)\ℍ has large quotients with only finitely many rational points. The results below were announced at the conference “Exponential sums” held in Jerusalem, January 1998. The complete proofs, which were presented in courses at Princeton (1997), are being prepared for publication.
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- [BS]A. Brumer and J. Silverman, The number of elliptic curves over Q with conductor N, preprint (1996).Google Scholar
- [By]V. Bykovsky, Trace formula for scalar product of Hecke series and its application (in Russian), preprint (1995)Google Scholar
- [DFI2]W. Duke, J. Friedlander and H. Iwaniec, Representations by the determinant and mean values of L-functions, in SieveMethods, Exponential Sums, and Their Applications in Number Theory (G. Greaves, G. Harman and M. Huxley, eds.), Cambridge University Press, Cambridge, 1997, pp. 109–115.CrossRefGoogle Scholar
- [Fr]J. B. Friedlander, Bounds for L-functions, in Proceedings of ICM 1994, Birkhäuser, Basel, 1995, pp. 363–373.Google Scholar
- [Ha]J. Hafner, On the zeros (à la Selberg) of Dirichlet series attached to certain cusp forms, in Topics in Analytic Number Theory (G. Kolesnik and J. Vaaler, eds.), University of Texas Press, Austin, 1985, pp. 125–164.Google Scholar
- [IS]H. Iwaniec and P. Sarnak, Dirichlet L-functions at the central point, in Number Theory in Progress (K. Györy, H. Iwaniec and J. Urbanowicz, eds.), Proceedings of the International Conference on Number Theory organized by the Stefan Banach International Mathematical Center in Honor of the 60th Birthday of Andrzej Schinzel, Walter deGruyter Mathematics/Mathematik, 1999, pp. 941–952.Google Scholar
- [KM]E. Kowalski and P. Michel, Sur le rang de J0(q), preprint (1997).Google Scholar
- [Se]A. Selberg, On the zeros of Riemann’s zeta-function, Collected Papers, Vol. 1, Springer-Verlag, Berlin, 1989, pp. 85–141.Google Scholar
- [Sh2]G. Shimura, in Introduction to the Arithmetic Theory of Automorphic Functions (Iwanomi Shoten, ed.), Princeton University Press, Princeton, NJ, 1971.Google Scholar
- [Va]J. M. Vanderkam, The rank of quotients of J0(N), preprint (1997).Google Scholar